Characterization of lenses

ABSTRACT

In accordance with one embodiment of the present invention, a method of characterizing a lens is provided. According to the method, an optical source such as a laser is configured to generate a collimated beam that is focused along an optical axis at a distance f ext . A test lens is placed along the optical axis, wherein the test lens is characterized by an effective focal length f i  that is substantially independent of incident irradiance. An output beam generated from the focused optical source and the test lens defines an output intensity profile at an observation plane located a distance Z 0  from the focal point of the optical source. The on-axis intensity I of the output intensity profile along the optical axis at the observation plane is monitored as the placement of the test lens along the optical axis is varied. A z-scan signature of the test lens is generated from the monitored intensity I. The z-scan signature is a plot of intensity as a function of Δz, where Δz represents a distance between a position of the test lens and the focal point f 0  along the optical axis. The effective focal length f i  of the test lens is characterized by determining the slope m 0  of the z-scan signature at Δz=0. Additional embodiments are disclosed and claimed.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application Ser.No. 60/676,499 (UVD 0343 MA), filed Apr. 29, 2005.

BRIEF SUMMARY OF THE INVENTION

The present invention relates to the characterization of lenses and,more particularly, to a scheme for characterizing the effective focallength, phase profile, and aberration parameters of a lens or lensarray. The methods of the present invention are applicable for any lensor lens array, whether it is an adaptive, variable focal length lens ora fixed focal length lens. In addition, it is contemplated that thepresent invention is applicable for short and long focal length lenses.However, it is particularly well suited for short focal length, adaptivelenses and adaptive electro-optic lenses, because these lenses aretypically difficult to characterize.

In accordance with one embodiment of the present invention, a method ofcharacterizing a lens is provided. According to the method, an opticalsource such as a laser is configured to generate a collimated beam whichis then focused along an optical axis at a distance f_(ext). A test lensis placed along the optical axis, wherein the test lens is characterizedby an effective focal length f_(i) that is substantially independent ofincident irradiance. An output beam generated from the focused opticalsource and the test lens defines an output intensity profile at anobservation plane located a distance Z₀ from the focal point f₀ of theoptical source. The on-axis intensity I of the output intensity profilealong the optical axis at the observation plane is monitored as theplacement of the test lens along the optical axis is varied. A z-scansignature of the test lens is generated from the monitored intensity I.The z-scan signature is a plot of intensity as a function of Δz, whereΔz represents a distance between a position of the test lens and thefocal point f₀ along the optical axis. The focal length f_(i) of thetest lens can be characterized by determining the slope$m_{0} = \frac{f_{ext}^{2}}{w_{0}^{2}z_{0}^{2}f_{i}}$of the z-scan signature at Δz=0, derived using ray optics.

In accordance with another embodiment of the present invention, thefocal length f_(i) of the test lens is characterized by determining aposition of the irradiance peak of the z-scan signature along Δz.

In accordance with another embodiment of the present invention, the testlens is characterized using the following formula:${{I\left( {0,0,Z_{0}} \right)} = {{f\left( {\Delta\quad z} \right)}\left( {1 + {\exp\frac{{- k_{0}}r_{0}^{2}b}{{q_{z}}^{2}}} - {2\exp\frac{{- k_{0}}r_{0}^{2}b}{2{q_{z}}^{2}}\cos\frac{{- k_{0}}r_{0}^{2}a}{2{q_{z}}^{2}}}} \right)}},$where I(0,0,Z₀) corresponds to the on-axis intensity and${{f\left( {\Delta\quad z} \right)} = {\frac{k_{0}^{2}}{4\pi^{2}Z_{0}^{2}f_{i}^{2}}\frac{{q_{0}}^{2}}{{q_{0}}^{2} + \left( {\Delta\quad z} \right)^{2}}{q_{z}}^{2}}},$a,b are the real and imaginary parts of the complex quantity q_(z),defined as${\frac{1}{q_{z}} = {\frac{1}{q_{0} + {\Delta\quad z}} + \frac{1}{Z_{0}} - \frac{1}{f_{i}}}},{q_{0} = {{{jk}_{0}{w_{0}^{2}/2}} = {j\frac{4f_{ext}^{2}}{k_{0}W_{0}^{2}}}}},{j = \sqrt{- 1}},$and k₀=2π/λ, where λ is the wavelength of the light source. This formulaalso incorporates the finite aperture size of the test lens 2r₀ and isderived using rigorous wave optics.

Accordingly, it is an object of the present invention to provide forimproved characterization of lenses. Other objects of the presentinvention will be apparent in light of the description of the inventionembodied herein.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The following detailed description of specific embodiments of thepresent invention can be best understood when read in conjunction withthe following drawings, where like structure is indicated with likereference numerals and in which:

FIG. 1 is a schematic illustration of a method of characterizing a lensaccording to the present invention;

FIG. 2 is a graph illustrating a two-dimensional z-scan signaturesaccording to the present invention; and

FIG. 3 is a graph illustrating slope values associated with the z-scansignatures of FIG. 2.

DETAILED DESCRIPTION

A method of characterizing a lens according to one embodiment of thepresent invention can be described with reference to FIG. 1, where anoptical source 10, a converging laser beam 20, and a test lens 30 areillustrated.

The optical source 10 includes a laser source (not shown) and lenselements arranged to generate a collimated optical beam that convergesas a focused laser beam 20 along an optical axis A at a distance f_(ext)behind a focusing lens of the optical source 10. The test lens 30 isalso placed along the optical axis A and may be characterized in termsof an unknown focal length f_(i). The test lens 30 is typically not anon-linear optical element and, as such, its focal length is independentof incident irradiance, i.e., the focal length will not vary to asignificant extent as the intensity of the incident laser beam varies.The focal length f_(ext) is illustrated in FIG. 1 as extending from theconverging lens of the optical source 10 to the focal point f₀. However,it is noted that the focal length f_(ext) will not correspond to thedistance between the converging lens and the focal point f₀ where thebeam incident upon the converging lens is not collimated.

An output beam 40 generated from the optical source 10 and the test lens30 defines an output intensity profile at an observation plane 50located a distance Z₀ from the focal point f₀. According to themethodology of one embodiment of the present invention, the intensity Iof the output beam 40 along the optical axis A at the observation plane50 is monitored as the placement of the test lens along the optical axisis varied and a z-scan signature of the test lens 30 is generated fromthe monitored intensity L

A z-scan signature, examples of which are illustrated in FIG. 2, is aplot of intensity I as a function of Δz, where Δz represents a distancebetween a position of the test lens and the focal point f₀ along theoptical axis A. The focal length f_(i) of the test lens 30 may becharacterized by determining the slope m₀ of the z-scan signature atΔz=0. The focal length f_(i) of the test lens can be characterized onthe premise that it is proportional to the slope at Δz=0. Morespecifically, the slope m₀ is used to characterize the focal lengthf_(i) of the test lens as follows:$m_{0} = \frac{f_{ext}^{2}}{w_{0}^{2}z_{0}^{2}f_{i}}$where f_(ext) is the focal length of the optical source, W₀ is one-halfthe initial width of the collimated laser beam 10, and Z₀ is a distancealong the optical axis A from the focal point f₀ to the observationplane 50.

The sign of the slope m₀, i.e., positive or negative slope, can be usedto characterize the test lens 30 as a positive or negative lens. Morespecifically, a positive slope at Δz=0 can be correlated with a positivefocal length f_(i) and a negative slope at Δz=0 can be correlated with anegative focal length f_(i).

In FIG. 2, respective z-scan signatures for eight different lenses areillustrated, i.e., 2 mm, 4 mm, 6 mm, and 8 mm positive and negativefocal length lenses. The signatures with intensity peaks on the negativeportion of the Δz axis correspond to negative focal length lenses whilethe intensity peaks on the positive portion of the Δz axis correspond topositive focal length lenses. FIG. 3 illustrates slope values associatedwith the respective z-scan signatures of FIG. 2, where the negativefocal length lenses are characterized by a negative slope at Δz=0 andthe positive focal length lenses are characterized by a positive slopeat Δz=0. As is illustrated in FIGS. 2 and 3, longer focal length lensesare characterized by smaller slopes at Δz=0.

According to one aspect of the present invention, the collimated beam ofthe optical source 10 is configured such that the converging laser beam20 approximates a converging Gaussian intensity profile. Typically,collimation is achieved using a laser source and a collimating lens. Thecollimated beam of the optical source 10 can be configured to generate aconverging laser beam 20 defined by a beam waist 2w₀ that is eithergreater or less than the aperture 2r₀ of the test lens 30. Beyond thesepoints, the particular configuration of the optical source 10 is not thesubject of the present invention and those practicing the presentinvention are directed to other resources for more specific instructionsregarding suitable optical source design.

According to one embodiment of the present invention, the test lens 30comprises a microlens array and the collimated beam of the opticalsource 10 is configured to generate a converging Gaussian laser beam 20defined by a beam width that may be greater than the respectiveapertures of the lens elements within the array over the scan length Δz.It is contemplated that the test lens may comprise a fixed focal lengthlens or a variable focal length lens comprising, for example, anelectrooptic element. In the case of the variable focal length lens, thez-scan signature can additionally be generated as a function of voltageapplied to the lens to generate a three-dimensional z-scan signaturecomprising dimensions Δz, intensity, and applied voltage. In addition,it is contemplated that the on-axis intensity I can be monitored as theplacement of the test lens 30 along the optical axis A is varied betweenpoints on opposite sides of the focal point f₀ or on a common side ofthe focal point f₀.

It is contemplated that particular sources of error in determining thefocal length f_(i) of the test lens 30 may be remedied, at least inpart, by ensuring that the observation plane 50 is positioned wellbeyond the focal point f₀. Under particular circumstances, it may bepreferable to position the observation plane sufficiently beyond thefocal point f₀ to approximate an infinitely displaced observation plane,where far-field approximations in wave optics can be used.

Another embodiment of the present invention contemplatescharacterization of the focal length f_(i) of the test lens 30 byidentifying an irradiance peak in the z-scan signature and determiningthe position of the irradiance peak along Δz. Specifically, referring tothe graph of FIG. 2, each of the z-scan signatures illustrated thereininclude an irradiance peak that is displaced to the right or left ofΔz=0. According to the contemplated method, the focal length f_(i) ofthe test lens 30 is presumed to be proportional to the value of Δzcorresponding to the irradiance peak. A positive value of Δz at theirradiance peak corresponds to a positive focal length f_(i) and anegative value of Δz at the irradiance peak corresponds to a negativefocal length f_(i).

Referring further to FIG. 2, it is noted that longer focal lengthdeterminations are prone to decreased accuracy when utilizing theirradiance peak of the z-scan to characterize the focal length f_(i) ofthe test lens 30. It is contemplated that it may be preferable to setthe observation plane 50 at almost infinity and subtract the value of Δzcorresponding to the irradiance peak from the observation distance toaddress this source of error. This shift in the position of theobservation plane 50 will shift the value of Δz corresponding to theirradiance peak to a slightly larger value, accounting for a significantportion of the calculation error.

Another embodiment of the present invention contemplatescharacterization of the test lens 30 by using the following formula:${{I\left( {0,0,Z_{0}} \right)} = {{f\left( {\Delta\quad z} \right)}\left( {1 + {\exp\frac{{- k_{0}}r_{0}^{2}b}{{q_{z}}^{2}}} - {2\exp\frac{{- k_{0}}r_{0}^{2}b}{2{q_{z}}^{2}}\cos\frac{k_{0}r_{0}^{2}a}{2{q_{z}}^{2}}}} \right)}},$where I(0,0,Z₀) corresponds to the on-axis intensity and${{f\left( {\Delta\quad z} \right)} = {\frac{k_{0}^{2}}{4\pi^{2}Z_{0}^{2}f^{2}}\frac{{q_{0}}^{2}}{{q_{0}}^{2} + \left( {\Delta\quad z} \right)^{2}}{q_{z}}^{2}}},$a,b are the real and imaginary parts of the complex quantity q_(z),defined as${\frac{1}{q_{z}} = {\frac{1}{q_{0} + {\Delta\quad z}} + \frac{1}{Z_{0}} - \frac{1}{f_{i}}}},{q_{0} = {{{jk}_{0}{w_{0}^{2}/2}} = {j\frac{4f_{ext}^{2}}{k_{0}W_{0}^{2}}}}},{j = \sqrt{- 1}},$and k₀=2π/λ, where λ is the wavelength of the light source. This formulaalso incorporates the finite aperture size of the test lens 2r₀ and isderived using rigorous wave optics.

More specifically, referring to FIG. 1, the optical field of a Gaussianbeam propagating in the Z-axis in the q-formalism has the general formof${E\left( {x,y,z} \right)} = {{E_{0}\frac{q_{0}}{q(z)}\exp} - {j\frac{\left( {x^{2} + y^{2}} \right)}{2{q(z)}}{\exp\left( {{- j}\quad k_{0}z} \right)}}}$with the beam width at a distance z, w(z), expressed as${w^{2}(z)} = {w_{0}^{2}\left( {1 + \left( \frac{z}{z_{R}} \right)^{2}} \right)}$and its radius of curvature R(z) as${R(z)} = {z\left( {1 + \left( \frac{z_{R}}{z} \right)^{2}} \right)}$where w₀ is the minimum beam width or waist at f_(ext), and z_(R) is theRaleigh length. The parameter z₀ is given by z_(R)=k₀w₀ ²/2, and k₀ isthe wave number given by k₀=2π/λ, where λ is the wavelength of the lightsource.

The test lens 30 with focal length f_(i) and aperture size of radius r₀,is placed around the focal point of a converging Gaussian beam, wherethe beam has a plane wavefront. In the two steps test, first the lensletwill be scanned for ±Δz distance about the focal point. The scanned Δzrange will be chosen to be a multiple of the Raleigh length, and suchthat the beam width at w(z) at a distance Δz is smaller than the lensaperture r₀. For a scanning distance Δz=nz_(R), where n is an integer,we have w(z)≈nw₀.

For example, for an original beam diameter at the focal point w₀=10 μm,the minimum lens aperture that can be used if we would like to scan forΔz=5z_(R) will be 50 μm. By plotting the on-axis intensity versus thescanned distance Δz, we can infer the focal length of the micro-lens. Ina variation of the test, the beam size may be larger than the size ofthe lenslet over the scan distance Δz. A diffraction pattern due to thepresence of the aperture will form. By changing the applied voltage thediffraction pattern shape will change. The focal length can be inferredfrom the peak and size of the inner lobe while the diffraction patternin general can produce information about the wave front aberration dueto the finite aperture size of the lenslet.

The Gaussian optical field distribution a distance Δz before the lensunder test is${E\left( {x,y} \right)} = {\frac{q_{0}}{q_{0} + {\Delta\quad z}}{\exp\left( {- \frac{j\left( {x^{2} + y^{2}} \right)k_{0}}{2\left( {q_{0} + {\Delta\quad z}} \right)}} \right)}}$where q_(o) is the q-parameter (=jk_(o)w_(o) ²/2) at the waist of thebeam w₀ at f_(ext).

Using the expression for E(x,y) above and the Fresnel Diffractionformula, one can express the optical field distribution on theobservation plane 50 in FIG. 1, taking into consideration the lensaperture, as${E\left( {x,y,Z_{0}} \right)} = {{\frac{j\quad k_{0}}{2\pi\quad Z_{0}}{\int{\int{{T\left( {x^{\prime},y^{\prime}} \right)} \times \exp}}}} - {\frac{j\quad k_{0}}{2Z_{0}}\left( {\left( {x - x^{\prime}} \right)^{2} + \left( {y - y^{\prime}} \right)^{2}} \right){\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}$where T(x′,y′)=E(x′,y′)×p_(f)(x′,y′)×T_(f)(x′,y′), and where it isassumed that Δz<<Z₀.

The preceding equation has 3-terms: (a) the Gaussian field distributionE(x,y); (b) the aperture function p_(f) (x,y)=circ(r/r₀)=1 for r<r₀,0otherwise; and (c) the lens phase function${T_{f}\left( {x^{\prime},y^{\prime}} \right)} = {\exp{\frac{{j\left( {x^{2} + y^{2}} \right)}k_{0}}{2f_{i}}.}}$After simple algebra, we have${E\left( {x,y,Z_{0}} \right)} = {C{\int{\int{{p_{f}\left( {x^{\prime},y^{\prime}} \right)}\exp\frac{- {jk}_{0}}{2Z_{0}}\left( {{xx}^{\prime} + {yy}^{\prime}} \right)\exp\frac{{- j}\quad k_{0}}{2q_{z}}\left( {x^{\prime 2} + y^{\prime 2}} \right){\mathbb{d}x^{\prime}}{\mathbb{d}y^{\prime}}}}}}$where$C = {{{\frac{j\quad k_{0}}{2\pi\quad Z_{0}} \times \frac{q_{0}}{q_{0} + {\Delta\quad z}}}\exp\frac{- {jk}_{0}}{2Z_{0}}\left( {x^{2} + y^{2}} \right)\quad{and}\quad\frac{1}{q_{z}}} = {\frac{1}{q_{0} + {\Delta\quad z}} + \frac{1}{Z_{0}} - {\frac{1}{f}.}}}$

The above-noted expression for E(x,y,Z₀) can be solved numerically toprovide the diffraction pattern at approximately a distance Z₀ from thelens under test. Such a diffraction pattern will be a function of thelens aperture and the applied voltage, since the lens focal lengthvaries with the applied voltage. The diffracted light intensity isI(x,y,Z₀)∝|E(x,y,Z₀)|².

It is contemplated that the test lens 30 may be further characterized byadding an aberration term to the formula for I(0,0,Z₀) and extractingaberration coefficients. Similarly, it is contemplated that the testlens 30 can be characterized by obtaining the focal length or the phaseprofile from the formula for I(0,0,Z₀). Where the phase profile isobtained, the lens may be characterized by comparing the phase profileof the test lens 30 with the ideal quadratic profile of the test lens 30to determine lens aberration.

It is noted that terms like “preferably,” “commonly,” and “typically”are not utilized herein to limit the scope of the claimed invention orto imply that certain features are critical, essential, or evenimportant to the structure or function of the claimed invention. Rather,these terms are merely intended to highlight alternative or additionalfeatures that may or may not be utilized in a particular embodiment ofthe present invention.

For the purposes of describing and defining the present invention it isnoted that the term “substantially” is utilized herein to represent theinherent degree of uncertainty that may be attributed to anyquantitative comparison, value, measurement, or other representation.The term “substantially” is also utilized herein to represent the degreeby which a quantitative representation may vary from a stated referencewithout resulting in a change in the basic function of the subjectmatter at issue.

Having described the invention in detail and by reference to specificembodiments thereof, it will be apparent that modifications andvariations are possible without departing from the scope of theinvention defined in the appended claims. More specifically, althoughsome aspects of the present invention are identified herein as preferredor particularly advantageous, it is contemplated that the presentinvention is not necessarily limited to these preferred aspects of theinvention.

1. A method of characterizing a lens, said method comprising:configuring an optical source to generate a laser beam that convergesalong an optical axis at a distance f_(ext) and corresponding focalpoint f₀; placing a test lens along said optical axis, wherein said testlens is characterized by an effective focal length f_(i) that issubstantially independent of incident irradiance; generating an outputbeam from said optical source and said test lens, said output beamdefining an output intensity profile at an observation plane located adistance Z₀ from the focal point f₀ defined by the optical source;monitoring on-axis intensity I of said output intensity profile alongsaid optical axis at said observation plane as the placement of saidtest lens along said optical axis is varied; generating a z-scansignature of said test lens from said monitored intensity I, where saidz-scan signature is a plot of intensity as a function of Δz, where Δzrepresents a distance between a position of said test lens and saidfocal point f₀ along said optical axis; and characterizing the focallength f_(i) of said test lens by determining the slope m₀ of saidz-scan signature at Δz=0. Focal Length f_(i)
 2. A method ofcharacterizing a lens as claimed in claim 1 wherein said focal lengthf_(i) of said test lens is characterized by calculating said focallength f_(i) as being proportional to a value of said slope at Δz=0. 3.A method of characterizing a lens as claimed in claim 1 wherein saidfocal length f_(i) of said test lens is characterized by correlating apositive slope at Δz=0 with a positive focal length f_(i) and a negativeslope at Δz=0 with a negative focal length f_(i).
 4. A method ofcharacterizing a lens as claimed in claim 1 wherein said focal lengthf_(i) of said test lens is characterized by determining a value and signof said slope at Δz=0.
 5. A method of characterizing a lens as claimedin claim 1 wherein said slope m₀ is used to characterize said focallength f_(i) of said test lens as follows:$m_{0} = \frac{f_{ext}^{2}}{w_{0}^{2}z_{0}^{2}f_{i}}$ where f_(ext) isthe focal length defined by said optical source, W₀ is one-half theinitial width of said converging laser beam generated by said opticalsource, and Z₀ is a distance along said optical axis from the focalpoint f₀ to the observation plane. Optical Beam Profile
 6. A method ofcharacterizing a lens as claimed in claim 1 wherein said converginglaser beam approximates a Gaussian intensity profile.
 7. A method ofcharacterizing a lens as claimed in claim 1 wherein said optical beamcomprises a collimated beam from a laser source and at least oneexternal lens positioned along said optical axis.
 8. A method ofcharacterizing a lens as claimed in claim 1 wherein said test lens has acharacteristic lens aperture and said optical beam width is eithergreater than or less than said characteristic lens aperture.
 9. A methodof characterizing a lens as claimed in claim 1 wherein: said test lenscomprises a microlens array; respective elements of said microlens arrayhave defined characteristic lens apertures; and said optical beam widthis either greater than or less than said characteristic lens apertures.Test Lens
 10. A method of characterizing a lens as claimed in claim 1wherein said test lens comprises a variable focal length electroopticlens.
 11. A method of characterizing a lens as claimed in claim 10wherein said z-scan signature is generated as a function of voltageapplied to said electrooptic lens to generate a three-dimensional z-scansignature comprising dimensions Δz, intensity, and applied voltage. 12.A method of characterizing a lens as claimed in claim 1 wherein saidtest lens comprises a microlens array. Observation Plane
 13. A method ofcharacterizing a lens as claimed in claim 1 wherein said observationplane is positioned beyond said focal point f₀.
 14. A method ofcharacterizing a lens as claimed in claim 13 wherein said observationplane is positioned sufficiently beyond said focal point f₀ toapproximate an infinitely displaced observation plane. Scanning
 15. Amethod of characterizing a lens as claimed in claim 1 wherein saidon-axis intensity is monitored as said placement of said test lens alongsaid optical axis is varied between points on opposite sides of saidfocal point or on a common side of said focal point.
 16. A method ofcharacterizing a lens, said method comprising: configuring an opticalsource to generate a laser beam that converges along an optical axis ata distance f_(ext) and corresponding focal point f₀; placing a test lensalong said optical axis, wherein said test lens is characterized by aneffective focal length f_(i) that is substantially independent ofincident irradiance; generating an output beam from said optical sourceand said test lens, said output beam defining an output intensityprofile at an observation plane located a distance Z₀ from the focalpoint f₀ defined by the optical source; monitoring on-axis intensity Iof said output intensity profile along said optical axis at saidobservation plane as the placement of said test lens along said opticalaxis is varied; generating a z-scan signature of said test lens fromsaid monitored intensity I, where said z-scan signature is a plot ofintensity as a function of Δz, where Δz represents a distance between aposition of said test lens and said focal point f₀ along said opticalaxis, and where said intensity plot comprises an irradiance peak alongΔz; and characterizing the focal length f_(i) of said test lens bydetermining a position of said irradiance peak along Δz.
 17. A method ofcharacterizing a lens as claimed in claim 16 wherein said focal lengthf_(i) of said test lens is proportional to the value of Δz correspondingto said irradiance peak.
 18. A method of characterizing a lens asclaimed in claim 16 wherein said signature is generated such that apositive value of Δz at said irradiance peak corresponds to a positivefocal length f_(i) and a negative value of Δz at said irradiance peakcorresponds to a negative focal length f_(i).
 19. A method ofcharacterizing a lens as claimed in claim 16 wherein said observationplane is positioned by subtracting the value of Δz corresponding to saidirradiance peak from an observation distance nearly equivalent to aninfinitely displaced observation plane.
 20. A method of characterizing alens, said method comprising: c configuring an optical source togenerate a laser beam that converges along an optical axis at a distancef_(ext) and corresponding focal point f₀; placing a test lens along saidoptical axis, wherein said test lens is characterized by an effectivefocal length f_(i) that is substantially independent of incidentirradiance; generating an output beam from said optical source and saidtest lens, said output beam defining an output intensity profile at anobservation plane located a distance Z₀ from the focal point f₀ definedby the optical source; measuring on-axis intensity I of said outputintensity profile along said optical axis at said observation plane;characterizing the test lens with finite aperture size 2r₀ using thefollowing formula${{I\left( {0,0,Z_{0}} \right)} = {{f\left( {\Delta\quad z} \right)}\left( {1 + {\exp\frac{{- k_{0}}r_{0}^{2}b}{{q_{z}}^{2}}} - {2\exp\frac{{- k_{0}}r_{0}^{2}b}{2{q_{z}}^{2}}\quad\cos\frac{k_{0}r_{0}^{2}a}{2{q_{z}}^{2}}}} \right)}},$where I(0,0,Z₀) corresponds to the on-axis intensity and${{f\left( {\Delta\quad z} \right)} = {\frac{k_{0}^{2}}{4\quad\pi^{2}Z^{2}f^{2}}\frac{{q_{0}}^{2}}{{q_{0}}^{2} + \left( {\Delta\quad z} \right)^{2}}{q_{z}}^{2}}},$a,b are the real and imaginary parts of the complex quantity q_(z),defined as${\frac{1}{q_{z}} = {\frac{1}{q_{0} + {\Delta\quad z}} + \frac{1}{Z_{0}} - \frac{1}{f_{i}}}},{q_{0} = {{{jk}_{0}{w_{0}^{2}/2}} = {j\frac{4f_{ext}^{2}}{k_{0}W_{0}^{2}}}}},{j = \sqrt{- 1}},$and k₀=2π/λ, where λ is the wavelength of the light source.
 21. A methodof characterizing a lens as claimed in claim 20 wherein said lens isfurther characterized by adding an aberration term to the formula forI(0,0,Z₀) and extracting aberration coefficients.
 22. A method ofcharacterizing a lens as claimed in claim 20 wherein said lens ischaracterized by obtaining the focal length from the formula forI(0,0,Z₀).
 23. A method of characterizing a lens as claimed in claim 20wherein said lens is characterized by obtaining the phase profile of thetest lens from the formula for I(0,0,Z₀).
 24. A method of characterizinga lens as claimed in claim 23 wherein said lens is characterized bycomparing the phase profile of the test lens with the ideal quadraticprofile of the test lens to determine lens aberration.